Mum's the word is a popular English idiom. It is related to an expression used by William Shakespeare, in Henry VI, Part 2.
"Mum's the word" means to keep silent or quiet. "Mum" is a Middle English word meaning "silent". The word may be derived from the "mummer" who does pantomime and just acts without saying anything. Note the phonetically similar German word "stumm" (Old High German "stum", Latin "mutus") meaning "silent, mute".
The origins of the phrase can be traced back to the fourteenth century and William Langland's narrative poem, Piers Plowman:
Then geten a mom of heore mouth til moneye weore schewed!
It can also be seen in popular fifteenth century Towneley Plays
and notably in Shakespeare's Henry VI, Part 2, Act 1, Scene 2: